This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
Simulation of MTBF with controls   F(t) = 1 - e ^ -λt   Where    • F(t) is the probability of failure    • λ is the failure rate in 1/time unit (1/h, for example)   • t is the observed service life (h, for example)  The inverse curve is the trust time On the right the increase in failures brings its
Simulation of MTBF with controls

F(t) = 1 - e ^ -λt 
Where  
• F(t) is the probability of failure  
• λ is the failure rate in 1/time unit (1/h, for example) 
• t is the observed service life (h, for example)

The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.

This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime, 
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)

Early Life
If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.

Useful Life
The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.  

Wearout
The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period. 
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
A model of the exponential growth phase of  E. coli  growth.
A model of the exponential growth phase of E. coli growth.
5 months ago
 ​Physical meaning of the equations  The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:        1. The prey population finds ample food at all times.    2. The food supply of the predator population depends entirely on the
​Physical meaning of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dtαx - βxy
 The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). {\displaystyle \displaystyle \gamma y} represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


Golbeter's model describes the circadian cycle of the fruitfly Drosophila     Goldbeter, Albert. "A model for circadian oscillations in the Drosophila period protein (PER)."  Proceedings of the Royal Society of London B: Biological Sciences  261.1362 (1995): 319-324                 .‏
Golbeter's model describes the circadian cycle of the fruitfly Drosophila 

Goldbeter, Albert. "A model for circadian oscillations in the Drosophila period protein (PER)." Proceedings of the Royal Society of London B: Biological Sciences 261.1362 (1995): 319-324





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This is a simulation that represents the populations of lions in the world over the last 200 years.
This is a simulation that represents the populations of lions in the world over the last 200 years.
Stock flow map of Immune System Cell type proliferation and differentiation, migration and circulation from Figs 1.3, 8.14 and 8.33 from Janeway's Immunobiology see  Insight  . Does not include cell death, interactions or inhibitory and promoting factors
Stock flow map of Immune System Cell type proliferation and differentiation, migration and circulation from Figs 1.3, 8.14 and 8.33 from Janeway's Immunobiology see Insight . Does not include cell death, interactions or inhibitory and promoting factors
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem. Also factored in is soil OM and solar radiation.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem. Also factored in is soil OM and solar radiation.
    Dynamic simulation modelers are particularly interested in understanding and being able to distinguish between the behavior of stocks and flows that result from internal interactions and those that result from external forces acting on a system.  For some time modelers have been particularly int

Dynamic simulation modelers are particularly interested in understanding and being able to distinguish between the behavior of stocks and flows that result from internal interactions and those that result from external forces acting on a system.  For some time modelers have been particularly interested in internal interactions that result in stable oscillations in the absence of any external forces acting on a system.  The model in this last scenario was independently developed by Alfred Lotka (1924) and Vito Volterra (1926).  Lotka was interested in understanding internal dynamics that might explain oscillations in moth and butterfly populations and the parasitoids that attack them.  Volterra was interested in explaining an increase in coastal populations of predatory fish and a decrease in their prey that was observed during World War I when human fishing pressures on the predator species declined.  Both discovered that a relatively simple model is capable of producing the cyclical behaviors they observed.  Since that time, several researchers have been able to reproduce the modeling dynamics in simple experimental systems consisting of only predators and prey.  It is now generally recognized that the model world that Lotka and Volterra produced is too simple to explain the complexity of most and predator-prey dynamics in nature.  And yet, the model significantly advanced our understanding of the critical role of feedback in predator-prey interactions and in feeding relationships that result in community dynamics.The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.
5. Predators have limitless appetite.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

Prey
When multiplied out, the prey equation becomes
dx/dtαx - βxy
 The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). {\displaystyle \displaystyle \gamma y} represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.


This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.
This simulation shows how plant, deer and wolf populations impact each other in a deciduous forest ecosystem.